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Solution of Linear Equation Systems

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Computational Methods for Fluid Dynamics

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Abstract

In this chapter, methods of solving the algebraic equation systems resulting from discretization of transport equations are described. Direct methods are briefly described, but the major part of the chapter is devoted to iterative solution techniques. Incomplete lower-upper decomposition, conjugate gradient and multigrid methods are given special attention. Approaches to solving coupled and non-linear systems are also described, including the issues of under-relaxation and convergence criteria. Various solvers can be downloaded from the book web-site (www.cfd-peric.de).

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Notes

  1. 1.

    Bini et al. (2009) review the history, extensions, and new proofs and formulas for cyclic reduction. It is being used as a smoother for multigrid applications on highly parallel, graphics processors (GPUs) now being used for fluid flow calculations (Göddeke and Strzodka 2011).

  2. 2.

    For example, \({\varvec{ \phi }}_3={\varvec{ \phi }}_0+{\varvec{ \rho }}^0+(I-A){\varvec{ \rho }}^0+ (I-A)(I-A){\varvec{ \rho }}^0\) .

  3. 3.

    The discussion in this book covers linear systems. Chapter 14 of Shewchuk (1994) describes the nonlinear conjugate gradient method and preconditioning for it.

  4. 4.

    For a simple rectangular geometry and Dirichlet boundary conditions (Brazier 1974): \(\omega =2/(1+\mathrm{sin}(\pi / N_{CV}))\).

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Stanford University, Stanford, CA, USA

    Joel H. Ferziger

  2. University of Duisburg-Essen, Duisburg, Germany

    Milovan Perić

  3. Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, USA

    Robert L. Street

Authors
  1. Joel H. Ferziger
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  2. Milovan Perić
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  3. Robert L. Street
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Correspondence to Milovan Perić .

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Ferziger, J.H., Perić, M., Street, R.L. (2020). Solution of Linear Equation Systems. In: Computational Methods for Fluid Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-99693-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-99693-6_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-99691-2

  • Online ISBN: 978-3-319-99693-6

  • eBook Packages: EngineeringEngineering (R0)

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